A Controlled Search for Radioactive Point Sources Branko Ristic
Mark Morelande
Ajith Gunatilaka
ISR division DSTO Melbourne VIC 3207 Australia
Melbourne Systems Lab The University of Melbourne Melbourne VIC 3010 Australia
HPP division DSTO Melbourne VIC 3207 Australia
[email protected]
[email protected]
[email protected]
Abstract—Given a map of a polygon-shaped search area with obstacles and a group of mobile networked observers equipped with radiation dose counters (such as the Geiger-Muller ¨ counter), the data fusion problem is twofold: (1) to establish if any radioactive point sources are present in the area; (2) if present, to determine their number and their parameters (locations and intensities of radiation). The detection/estimation part of the problem is solved in the Bayesian framework using a particle filter. The control of multiple observers during the search is carried out by maximization of the information gain measured by the R´enyi divergence.
Keywords: Nuclear search, obstacles, particle filter, sensor management, search strategy, CBRN data fusion I. I NTRODUCTION Detection and localisation of nuclear radioactive materials is of growing concern for homeland security. Numerous cases of lost or stolen radioactive materials have been reported [1], [2]. The danger is that these materials could be used by the terrorists to build improvised nuclear devices and deploy them in densely populated areas to cause destruction and panic. The problem of nuclear material detection, localisation and search has been considered in a number of recent publications, see for example [3]–[8]. The authors have earlier contributed to this growing research field by deriving the Cramer-Rao lower bounds and proposing statistically/computationally efficient algorithms for detection and localisation of multiple static radioactive sources [9]. The main difference in the present paper is that we allow the sensors to be controlled and thus we develop a search strategy. The adopted search strategy combines the simultaneous detection and localisation of point sources of gamma radiation with sensor control. The sources are assumed static, that is with constant intensity and no motion. The search is carried out by a team of coordinated multiple mobile observers. Each observer (a robot or a human) is equipped with: (1) a gamma radiation survey instrument (e.g. Geiger-M¨uller counter), measuring dose-counts per second; (2) a precise self-localisation instrument (e.g. differential GPS receiver); (3) a two-way radio link with the fusion and control center (FCC); (4) a computing device which serves as an interface with other units. Using the radio link, each observer sends occasionally to the FCC c
Commonwealth of Australia
a message which consist of: (i) a dose-count measurement; (ii) own position (no need to send the time stamps because the sources are static). The FCC is assumed to have a map of the search area and the knowledge of gamma-radiation attenuation factors for all obstacles present in the search area. The observer receives from the FCC the control (action) vectors, which specify the new observer location (where to go) and the radiation exposure time (how long to stay there). Clearly we assume here a centralized data fusion and control architecture. The goal of the search is to establish in a quick and safe manner how many nuclear sources are present in the designated search area and for each of the sources to estimate its location and intensity of radiation. Similar search strategies have been addressed in [10] and [11]. The present paper, however, incorporates three novelties: the search is carried out in the presence of gammaattenuating obstacles; we search for multiple gamma-radiation point sources (rather than a single source); the particle filter incorporates the progressive correction technique which results in more reliable detection/estimation. The paper is organised as follows. A formal problem description is presented in Sec.II. The integrated detector/estimator and its particle filter based implementation are described in Sec.III. The observer control is discussed in Sec.IV, the handling of irregular search area in Sec. V. The simulation results are presented in Sec.VI, and the paper is summarised in Sec.VII. II. P ROBLEM
DESCRIPTION
The radiation counts from nuclear decay obey Poisson statistics [12], [13], experimentally verified in [14]: the probability that a gamma radiation detector registers z ∈ Z+ counts in τ seconds, from a source that emits on average µ counts per second is: λz −λ e , (1) P(z; λ) = z! where λ = µτ is the parameter (the mean and the variance) of the Poisson distribution. A radiation source is parametrised by • Its location (xs , ys ) in the Cartesian coordinate system, such that (xs , ys ) ∈ A; • Its equivalent intensity rate, Is , which is a single parameter that takes into account the activity of the radioactive
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source, the value of gamma energy per disintegration and the scaling factors involved [15]. The corresponding parameter vector of source r = 1, 2, . . . , R is assumed to be time-invariant and thus given by: ⊺ xr = xr yr Ir , (2)
where ⊺ denotes the matrix transpose. The parameter vector in the presence of R > 0 sources is then defined as a concatenated vector: X(R) = [x⊺1
...
x⊺R ]⊺ .
(3)
In the case R = 0 (no sources present), by convention X(0) = ∅. Given the concatenated parameter vector X(R) (and thus implicitly given the value of R = dim[X(R)]/3), the likelihood function of a radiation count measurement zki , registered at precisely known location (ξki , ζki ) by observer i ∈ {1, . . . , S} at time tk is given by: (4) p(zki |X(R)) = P zki ; λik (X(R))
where P is the Poisson distribution defined in (1). Parameter λik (X(R)) in (4) is the mean radiation count which can be expressed as follows: M rki P I exp − φ ∆ R r mrki mrki X i m=1 p λik (X(R)) = µb + τk i i 2 2 (ξk − xr ) + (ζk − yr ) r=1
(5) where i • τk is the radiation exposure interval of sensor i (i.e. measurement zki is collected over the time interval [tk − τki , tk ]); • µb represents the average count-rate due to the background radiation (assumed known). • The exponential term on the right-hand side of (5) is due to the gamma-ray attenuation of obstacles; Mrki is the number of obstacles between observer i and source r at time k, φmrki is the attenuation factor of obstacles m = 1, . . . , Mrki (φmrki is assumed known) and ∆mrki is the thickness of obstacle m along the ray path between source r and observer i. By controlling the motion and exposure intervals of the observers, the goal of the search algorithm is to determine in the quickest possible manner the number of radioactive sources, their locations and intensities. III. BAYESIAN S OURCE D ETECTION
AND
E STIMATION
The observers operate asynchronously and the discrete-time index k in this case refers to the sequence number of a message sent by observer i, containing information on the count-measurement zki and observer own-position (ζki , ηki ). For simplicity and without loss of generality, in this section we can drop the superscript i from notation. The choice of the measurement location (ζki , ηki ) and the exposure time τki will be discussed in Sec. IV.
In order to estimate the number of sources (detection part of the problem), we treat parameter R as a discretevalued random variable R ∈ {0, 1, . . . , Rmax }. Joint detection and estimation is then cast as a hybrid (continuous-discrete) estimation problem and solved within the sequential Bayesian framework [16, Ch.11], [17]. Let the posterior density of the parameter vector X(R), after processing k − 1 measurements, be denoted by p(X(R)|z1:k−1 ), where z1:k−1 = {z1 , . . . , zk−1 } is the accumulated set of dose-count measurements. Given a new kth measurement zk , the goal is to construct the updated posterior density p(X(R)|z1:k ) using the measurement likelihood. For the case R = r, where r = 1, . . . , Rmax , the update step is done via: p(zk |X(r)) p(X(r)|z1:k−1 ) p(X(r)|z1:k ) = (6) p(zk |z1:k−1 ) with the likelihood function p(zk |X(r)) introduced in (4) and (5). Once the posterior pdf p(X(r)|z1:k ) is known, the posterior probability Pk (r) = P r{R = r|z1:k } that r sources are present in A is computed as the marginal: Z Pk (R) = p(r = R, X(r)|z1:k )dX (7) From this, the MAP estimate of the number of sources after processing k measurements can be computed as ˆ k = arg R
max
0≤R≤Rmax
Pk (R)
(8)
Similarly, the posterior pdfs of individual source parameter ˆ k , can be computed as the vectors xr , for r = 1, . . . , R ˆ marginals of pdf p(X(Rk )|z1:k ). The described hybrid state estimator is implemented in the form of a particle filter (PF). A particle set {wkn , Xnk (Rkn )}N n=1 is a random sample representation of the posterior pdf p(X(R)|z1:k ), with N being the number of particles and wkn the importance weight of particle n. Each particle is thus characterised by the number of radiological sources Rkn ∈ {0, . . . , Rmax } andh accordingly the concatenated parameter i⊺ vector Xnk (Rkn ) = (xn1,k )⊺ , . . . , (xnRn ,k )⊺ ; k The PF is initialised using prior knowledge. The prior probability on the source number is uniform, i.e. P0 (r) = 1/(1 + Rmax ), r = 0, . . . , Rmax . The source location prior is n uniform within the search region A, i.e. (xnr,0 , yr,k ) ∼ U(A). The prior pdf on source intensity is the Gamma density with n shape parameter κ and scale parameter β, Ir,0 ∼ Γ(κ, β). Assuming that resampling of particles has been carried out at time k − 1, then all importance weights at time k − 1 are n equal wk−1 = 1/N . Then the basic idea of the update step of the PF is to compute the new importance weights using measurement zk as follows: wkn = Cp(zk |Xnk (Rkn ))
(9)
where C is such that the weights wkn sum up to one, p(zk |Xnk (Rkn )) is the likelihood of particle n, see (4) and (5). However the straightforward application of (9), followed by
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resampling, can cause the particle filter to collapse for two reasons that lead to the particle degeneracy. First, the posterior at k − 1 can be expected to be much more diffuse than the likelihood. As a result, many (if not all) samples will be drawn in undesirable parts of the parameter space and consequently may be given zero weights. Second, since we consider a static parameter estimation case here, the variety of particles will decrease with discrete-time k because of the accumulation of the repetition in the sample and all particles may ultimately concentrate on the single point of the state-space. In order to overcome the described deficiencies of the straightforward implementation of the update step, we apply the progressive correction (PC) technique with particle regularisation [18]. In order to explain the PC technique, let us rewrite (6) in a shortened form as π = ℓπ0 /c, where ℓ is the likelihood function, π and π0 are the posterior and prior pdfs, respectively, and c is a normalising constant. Suppose we decompose the likelihood as a sequence (ℓ1 , ℓ2 , . . . , ℓS ) such that ℓ =
S Y
ℓs .
(10)
s=1
Observe that the update π = ℓπ0 /c is equivalent to the sequence of update steps: π = ℓS (ℓS−1 (. . . ℓ2 (ℓ1 π0 /c)))
(11)
The decomposition of the likelihood function (10) should be done in such a way that in the early stages (for small s) the importance weights are computed using ℓs which is somewhat more diffuse than the true likelihood ℓ. This can be achieved by raising the likelihood function at stage s = 1, . . . , S to the PSpower γs which is smaller than one: γs ∈ (0, 1] and s=1 γs = 1. Thus the likelihood used for s < S will be broader than the true likelihood, particularly in the earlier stages, making it more probable that samples drawn from the diffuse prior will have a high likelihood. In the later stages the likelihood approximation sharpens so that the samples gradually concentrate in the area of the parameter space suggested by the true likelihood. For the selection of coefficients γs we have used the adaptive scheme suggested in [18]. Note that after resampling we perform particle regularisation, both in the continuous domain (x, y, I) and in the discrete-domain R. In the discrete domain this means that we increment or decrement by 1 the value of Rkn with probability Pb (birth) or Pd (death), respectively. The particle filter outputs at each k the probabilities Pk (r) of (7), r = 1, . P . . , Rmax , approximated as: Pk (r) ≈ Mr /N , N n where Mr = n=1 δK [Rk , r], (δK [i, j] is the Kronecker delta) is the number of particles with r sources. The MAP estimate of R is thenPcomputed using (8), and the MMSE N estimate of R by 1/N n=1 Rkn . When the search is complete ˆ k > 0, the PF outputs the estimate of the concatenated and R parameter vector computed as the sample mean: ˆ k (R ˆk ) = X
N 1 X n ˆ k ]. X · δK [Rkn , R MRˆ k n=1 k
(12)
IV. O BSERVER CONTROL A. Control vectors In order to carry out the search mission in the most efficient manner, the observers are controlled by the FCC to take actions that will lead to the maximum information gain at each time step k. The information gain is measured using the current knowledge of the state, which is fairly poor initially but gradually improves with more measurements being available. After processing a message from observer i ∈ {1, . . . , S}, collected at time k, the FCC determines the control vector uik which is subsequently transmitted back to observer i. The control vector is three-dimensional and defined as: uik = ζki ′ ηki ′ τki ′ . (13)
where1 k ′ > k. Since the control space is continuous-valued, we discretise it. If the current position of observer i is (ξki , ζki ), the provisional set of one-step ahead admissible locations for observer i is selected as: n ξki + jρ0 cos(ℓθ0 ), ζki + jρ0 sin(ℓθ0 ) ; o j = 0, . . . , Nρ − 1; ℓ = 0, 1, . . . , Nθ − 1 (14) where θ0 = 2π/Nθ and ρ0 is a conveniently selected radial step. In this way the observer can stay in its current location (j = 0), or can move radially in incremental steps. The actual set of admissible locations is a subset of (14): it excludes all future locations (ξki ′ , ζki ′ ) for which the straight-line path from the current location steps outside the search region A. In order to reduce the dimension of the control vector optimisation space, for a given new prospective location (ξki ′ , ζki ′ ), the exposure time is computed directly using the maximum tolerable radiation exposure as a parameter. When the next course of action is decided, the controller considers the information gain for h ∈ N future steps ahead [19]. The case h = 1 is known as myopic sensor management: it is greedy in the sense that it controls the observers for immediate information gain. Very often, especially in the presence of obstacles, it is better to choose the action which will provide more information in the longer run (for h > 1). Based on (14), the number of possible control vectors to consider for h step ahead control is (Nρ Nθ )h ; it grows exponentially with h. In order to reduce this growth, we adopt a simple heuristic: for horizons h = 2, 3, . . . only one action is admissible, the same one that was carried out at h = 1. In this way for a hstep ahead control we need to consider only hNρ Nθ control vectors. B. Information gain In early attempts [10] we computed the information gain using the Fisher information. Since we deal here with a hybrid estimation problems, the use of the Fisher information (which is related only to the second order estimation error performance) is clearly inappropriate, due to the discrete-valued 1 We use k ′ rather than k + 1 here because the next (k + 1)st measurement in the sequence may arrive from a different observer.
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variable R. A much better solution in this case is to use an information gain formula which involves the entire probability density function. In this paper for this purpose we adopt the information gain based on the R´enyi (or alpha) divergence between two posterior densities. This approach will enable us to control the observers with the maximum information gain throughout the entire search duration, irrespective of the ˆk . current estimate of the number of sources R The R´enyi divergence between densities f0 and f1 is defined as [19], [20]: Z 1 ln f1α (x)f01−α (x)dx (15) Dα (f1 ||f0 ) = α−1
VI. S IMULATION
The irregular shape of the search area A is shown in Fig.1.(a). There are three obstacles: two are drawn using black lines and one is in green. The obstacles in black are transparent for gamma radiation from radioactive sources (no attenuation). The obstacle in green is characterised by attenuation factor φ = 0.05, see eq.(5). There are two sources in A, indicted by green asterisks, placed at (x1 = 120m, y1 = 300m) and (x2 = 650m, y2 = 400m) with intensities I1 = 20 · 103 cnt/s and I2 = 23 · 103 cnts/s, respectively.
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where the integral is of the dimension of the state space and α is a parameter which determines how much we emphasize the tails of two distributions in the metric. Our goal is to compute, as the measure of the information gain, the alpha-divergence between the posterior density at time k, f0 = p(X(R)|z1:k ), and the future posterior density, f1 = p(X(R)|z1:k , zkik′ ), computed using the new measurement zkik′ from sensor ik . After incorporating the Bayes rule and the particle approximation of posterior densities (after resampling, hence with equal importance weights), the expression for R´enyi divergence becomes [21]: PN ik n n α p(zk′ |Xk′ (Rk′ )) 1 iα Dα (f1 ||f0 ) ≈ ln hPn=1 N α−1 p(z ik |Xn (Rn )) n=1
k′
k′
RESULTS
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(16)
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p(zkik′ |Xnk′ (Rkn′ ))
The likelihood of particle n, was introduced in (4) and (5). Since the future measurement zkik′ , which is required in (16), is not available at the time of making the sensor control decision, we need to use the expected value of the alpha-divergence over all possible measurement values zkik′ . The search is terminated either if the total search time exceeds a given interval or if ∃r ∈ {0, . . . , Rmax } such that Pk (r) ≥ 1 − ǫ during a certain period of time and the spread of particles in x − y plane is below a certain threshold.
Exposure τ [s]
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T i m e [s] 2 1.5
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V. R AY
15
CASTING ALGORITHM
1 0.5
The search region A is a polygon with possibly polygon shaped obstacles (holes). This irregular shape of the search region affects both the implementation of the particle filter and the selection of control vectors. The key problem in placing the particles or selecting the observer action is to determine if a certain point or a line belongs to the interior of A. For this purpose we use the ray casting algorithm [22] which has been originally developed in computer graphics. To determine if a point (xp , yp ) belongs to the interior of a polygon with polygonal holes, we cast a horizontal ray emanating from (xp , yp ) to the right. If the number of times this ray intersects the line segments making up the polygon is an even number, then the point is outside the polygon, and vice versa.
0 0
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T i m e [s]
Figure 1. The search results after 140 seconds: (a) search area; (b) exposure intervals τki versus time; (c) The MMSE estimate of R versus time
Fig.1.(a) also shows the trajectories of two observers after 140 seconds of search. The pink coloured path corresponds to the observer that started from the position (350m, 750m). The cyan-coloured path, which started from (450m, 750m)), is considerably longer. This is due to the longer exposure intervals of the pink observer which consequently stays longer in measurement positions. The exposure intervals are shown in Fig.1.(b) and the MMSE estimate of Rk in Fig.1.(c), using
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the same colour coding. In the simulations we assume that the messages are sent to the FCC and the control vectors received (by observers) instantaneously. Every such event is indicated on the time axis in Fig.1.(b) and (c). Note how these events happen asynchronously, due to the different exposure intervals of observers. The cyan observer, as it approaches one of the sources, is exposed more to radiation and therefore is controlled to operate with shorter exposure intervals. The red dots in Fig.1.(a) indicate the locations of particles with Rkn = 1, while the blue dots indicate the positions of particles with Rkn = 2. Since the particles with Rkn = 2 contain two source states, there are two types of blue particles (tiny squares and circles). After 140 seconds none of the sources has been localised yet, although the MMSE estimate of Rk in Fig.1.(c) indicates the presence of more than one source. The parameters used by the PF in simulations are: µb = 1 cnt/s, Rmax = 2, N = 2000, h = 3, Nθ = 8, Nρ = 3, ρ0 = 15m, κ = 4, β = 5000, Pb = Pd = 0.003. The R´enyi divergence was computed with α = 0.5. The search terminated after 6 minutes and 20 seconds and resulted in both sources being detected and localised. The final source estimates were: x ˆ1 = 119.50m, yˆ1 = 297.42m, Iˆ1 = 19344 cnt/s for source 1 and x ˆ1 = 648.97m, yˆ1 = 399.28m, Iˆ1 = 21463 cnt/s for source 2. Table 1 shows the results from 20 Monte Carlo runs. In order to explain what is the difference between the detected and localised sources in the table, recall that if the two sources are not localised in a given time (in this case 15 minutes), the search is terminated. Using S = 1 observer, on one occasion (out of 20), only a single source has been localised in the given timeframe2; on all occasions the estimated number of sources was correct (detection part). The results of Table 1 essentially demonstrate the benefits of using more than one observer in the search: the localisation is more reliable and quicker. Table I S EARCH PERFORMANCE (20 RUNS )
Detected 2 sources : Localised 2 sources : RMS Error in x [m] RMS Error in y [m] RMS Error in I [cnt/s] Average search time [s]
one observer (S = 1) 20 19 8.34 4.82 3431 531
two observers (S = 2) 20 20 4.76 8.67 3606 384
VII. C ONCLUSIONS The paper presented a controlled search algorithm for detection and parameter estimation of an unknown number of point sources of gamma radiation. The search area can have an irregular shape with gamma radiation attenuating objects. The numerical results demonstrate excellent results. The future
work will study the sensitivity of the algorithm to the uncertain levels of background radiation µb . ACKNOWLEDGEMENTS The authors would like to thank Dr A. Skvortsov (DSTO) and Dr R. Gailis (DSTO) for useful technical discussions. R EFERENCES [1] W. K. H. Panofsky, “Nuclear proliferation risks, new and old,” Issues in Science and Technology, vol. 19, 2003, http://www.issues.org/19.4/panofsky.html. [2] IAEA, “Trafficking in nuclear and radioactive material in 2005,” Aug. 2006, http://www.iaea.org/NewsCenter/News/2006/traffickingstats2005.html. [3] J. W. Howse, L. O. Ticknor, and K. R. Muske, “Least squares estimation techniques for position tracking of radioactive sources,” Automatica, vol. 37, pp. 1727–1737, 2001. [4] D. M. Nicol, R. Tsang, H. Ammerlahn, and M. M. Johnson, “Sensor fusion algorithms for the detection of nuclear material at border crossings,” in Proc. SPIE, vol. 6201, 2006. [5] R. J. Nemzek, J. S. Dreicer, D. C. Torney, and T. T. Warnock, “Distributed sensor networks for detection of mobile radioactive sources,” IEEE Trans. Nuclear Science, vol. 51, no. 4, pp. 1693–1700, 2004. [6] S. M. Brennan, A. M. Mielke, and D. C. Torney, “Radioactive source detection by sensor networks,” IEEE Trans. Nuclear Science, vol. 52, no. 3, pp. 813–819, 2005. [7] D. L. Stephens and A. J. Peurrung, “Detection of moving radioactive sources using sensor networks,” IEEE Trans. Nuclear Science, vol. 51, no. 5, pp. 2273–2278, 2004. [8] A. Sundaresan, P. K. Varshney, and N. S. V. Rao, “Distributed detection of a nuclear radioactive source using fusion of correlated decisions,” in Proc. 10th Int. Conf. Information Fusion, Qu´ebec, Canada, July 2007. [9] M. Morelande, B. Ristic, and A. Gunatilaka, “Detection and parameter estimation of multiple radioactive sources,” in Proc. 10th Int. Conf. Information Fusion, Qu´ebec, Canada, July 2007. [10] B. Ristic and A. Gunatilaka, “Information driven localisation of a radiological point source,” Information Fusion, 2008, (In print, available on-line from ScienceDirect). [11] B. Ristic, M. Morelande, A. Gunatilaka, and M. Rutten, “Search for a radioactive sources: Coordinated multiple observers,” in Proc. 3rd IEEE Int. Conf. Intelligent Sensors, Sensor Netorks and Information Processing (ISSNIP), Melbourne, Dec. 2007. [12] N. Tsoulfanidis, Measurement and detection of radiation. Washington, DC : Taylor & Francis, 1995. [13] A. V. Klimenko, W. C. Priedhorsky, N. W. Hengartner, and K. N. Borozin, “Efficient strategies for low-level nuclear searches,” IEEE Trans. Nuclear Science, vol. 53, no. 3, pp. 1435–1442, June 2006. [14] A. Gunatilaka, B. Ristic, and R. Gailis, “On localisation of a radiological point source,” in Proc. Information, Decision and Control (IDC), Adelaide, Australia, Feb. 2007. [15] A. Martin and S. A. Harbison, An introduction to radiation protection. Chapman & Hall, 1987. [16] B. Ristic, S. Arulampalam, and N. Gordon, Beyond the Kalman filter: Particle filters for tracking applications. Artech House, 2004. [17] J. Czyz, B. Ristic, and B. Macq, “A particle filter for joint detection and tracking of color objects,” Image and Vision Computing, vol. 25, pp. 1271–1281, 2007. [18] C. Musso, N. Oudjane, and F. LeGland, “Improving regularised particle filters,” in Sequential Monte Carlo methods in Practice, A. Doucet, N. deFreitas, and N. J. Gordon, Eds. New York: Springer-Verlag, 2001. [19] A. O. Hero, D. Castanon, D. Cochran, and K. Kastella, Eds., Foundations and applications of sensor management. Springer, 2007. [20] C. M. Kreucher, A. O. Hero, K. D. Kastella, and M. R. Morelande, “An information based approach to sensor management in large dynamic networks,” Proc. of the IEEE, 2007. [21] C. M. Kreucher, K. D. Kastella, and A. O. Hero, “Sensor management using an active sensing approach,” Signal Processing, vol. 85, pp. 607– 624, 2007. [22] J. D. Foley, Computer Graphics: Principles and Practice. AddisonWesley, 1995.
2 The source is considered localised when the particle spread in the x − y plane is below a given threshold.
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